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If your definition of "something" is very weak, then yes. For example, none of the $p_i$ can appear in the prime factorization of $n+1$.

In some very simple cases, such as when $n$ is a perfect power, there may be algebraic identities that make your life easier. For example, if $n=k^3$, then $n+1=k^3+1=(k+1)(k^2-k+1)$. There's no guarantee that either of $k+1$ or $k^2-k+1$ is prime, but at least they're smaller than $n+1$: that is, you've gotten started...

Even in certain extraordinarily simple cases, though, it can be hard to get anywhere. For example, it's still an open problem whether $2^{2^k}+1$ is prime for infinitely many $k$ — and the numbers $2^{2^k}$ have about as simple a prime factorization as you could imagine!